232 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2? i.e., goods entering into the objective, and by maximizing a common scalar factor applied to these quantities (Figure 4). This problem can also be formulated in linear programming terms: One adds to the constraints (1) linear equalities expres- sing the prescribed ratios, and chooses as a maximand (2) the quantity of any one desired good, say. In convex programming the feasible set is defined by g(x, ..., x,)=20, 1=1, ..., m, where the g; are concave (1) functions, and the maximand U Ux, .... x, A $ concave function g(#y, ..., #,) is represented by a hypersurface £.%, ..., %,) in the space {y, x, ..., #,} that is never « below » any its chords (if the ++ direction is « up ») Koopmans - pag.